# Properties

 Label 2.152.6t5.a.b Dimension $2$ Group $S_3\times C_3$ Conductor $152$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3\times C_3$ Conductor: $$152$$$$\medspace = 2^{3} \cdot 19$$ Artin stem field: Galois closure of 6.0.184832.1 Galois orbit size: $2$ Smallest permutation container: $S_3\times C_3$ Parity: odd Determinant: 1.152.6t1.c.b Projective image: $S_3$ Projective stem field: Galois closure of 3.1.2888.1

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - 2x^{5} + x^{4} + 2x^{3} - 4x + 3$$ x^6 - 2*x^5 + x^4 + 2*x^3 - 4*x + 3 .

The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 7.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$: $$x^{2} + 6x + 3$$

Roots:
 $r_{ 1 }$ $=$ $$5 a + 4 + 4\cdot 7 + 2 a\cdot 7^{2} + \left(4 a + 6\right)\cdot 7^{3} + \left(a + 1\right)\cdot 7^{4} + \left(5 a + 5\right)\cdot 7^{5} + \left(6 a + 5\right)\cdot 7^{6} +O(7^{7})$$ 5*a + 4 + 4*7 + 2*a*7^2 + (4*a + 6)*7^3 + (a + 1)*7^4 + (5*a + 5)*7^5 + (6*a + 5)*7^6+O(7^7) $r_{ 2 }$ $=$ $$2 a + 2 + 6 a\cdot 7 + \left(4 a + 2\right)\cdot 7^{2} + \left(2 a + 1\right)\cdot 7^{3} + \left(5 a + 6\right)\cdot 7^{4} + \left(a + 1\right)\cdot 7^{5} +O(7^{7})$$ 2*a + 2 + 6*a*7 + (4*a + 2)*7^2 + (2*a + 1)*7^3 + (5*a + 6)*7^4 + (a + 1)*7^5+O(7^7) $r_{ 3 }$ $=$ $$6 a + \left(3 a + 1\right)\cdot 7 + 4\cdot 7^{2} + \left(a + 3\right)\cdot 7^{3} + \left(3 a + 2\right)\cdot 7^{4} + \left(3 a + 6\right)\cdot 7^{5} + 7^{6} +O(7^{7})$$ 6*a + (3*a + 1)*7 + 4*7^2 + (a + 3)*7^3 + (3*a + 2)*7^4 + (3*a + 6)*7^5 + 7^6+O(7^7) $r_{ 4 }$ $=$ $$a + 6 + \left(3 a + 5\right)\cdot 7 + 6 a\cdot 7^{2} + \left(5 a + 4\right)\cdot 7^{3} + \left(3 a + 4\right)\cdot 7^{4} + \left(3 a + 6\right)\cdot 7^{5} + \left(6 a + 5\right)\cdot 7^{6} +O(7^{7})$$ a + 6 + (3*a + 5)*7 + 6*a*7^2 + (5*a + 4)*7^3 + (3*a + 4)*7^4 + (3*a + 6)*7^5 + (6*a + 5)*7^6+O(7^7) $r_{ 5 }$ $=$ $$4 a + \left(4 a + 1\right)\cdot 7 + \left(2 a + 4\right)\cdot 7^{2} + \left(5 a + 1\right)\cdot 7^{3} + \left(4 a + 3\right)\cdot 7^{4} + \left(a + 5\right)\cdot 7^{5} +O(7^{7})$$ 4*a + (4*a + 1)*7 + (2*a + 4)*7^2 + (5*a + 1)*7^3 + (4*a + 3)*7^4 + (a + 5)*7^5+O(7^7) $r_{ 6 }$ $=$ $$3 a + 4 + \left(2 a + 1\right)\cdot 7 + \left(4 a + 2\right)\cdot 7^{2} + \left(a + 4\right)\cdot 7^{3} + \left(2 a + 2\right)\cdot 7^{4} + \left(5 a + 2\right)\cdot 7^{5} + \left(6 a + 6\right)\cdot 7^{6} +O(7^{7})$$ 3*a + 4 + (2*a + 1)*7 + (4*a + 2)*7^2 + (a + 4)*7^3 + (2*a + 2)*7^4 + (5*a + 2)*7^5 + (6*a + 6)*7^6+O(7^7)

## Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

 Cycle notation $(2,5,4)$ $(1,6,3)$ $(1,4)(2,6)(3,5)$

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character value $1$ $1$ $()$ $2$ $3$ $2$ $(1,4)(2,6)(3,5)$ $0$ $1$ $3$ $(1,6,3)(2,5,4)$ $-2 \zeta_{3} - 2$ $1$ $3$ $(1,3,6)(2,4,5)$ $2 \zeta_{3}$ $2$ $3$ $(1,6,3)$ $-\zeta_{3}$ $2$ $3$ $(1,3,6)$ $\zeta_{3} + 1$ $2$ $3$ $(1,3,6)(2,5,4)$ $-1$ $3$ $6$ $(1,2,6,5,3,4)$ $0$ $3$ $6$ $(1,4,3,5,6,2)$ $0$

The blue line marks the conjugacy class containing complex conjugation.